There's a ton of good information here, but it seems to assume a guitar or piano, where there's only one key or fretted space for each note.

For a fretless stringed instrument, they are indeed different notes, and the same note within a single piece can sound different depending on whether the line is moving up or down.

If that sounds heretical: I got this from the Alexander String Quartet, in the Q&A session after their performance. They have a measurement of microtones (I think they're called "clicks" but I forget), and all four of them have to agree on how many clicks up or down from the center of the note they're using.